The Dirac equation in a Yang-Mills field as an equation for just one real function

TL;DR

This paper extends previous work showing the Dirac equation in an electromagnetic field can be reduced to a single real function, now applying the same reduction to the more complex Yang-Mills field.

Contribution

It demonstrates that the Dirac equation in a Yang-Mills field can also be expressed as a fourth-order equation for one real function, generalizing earlier electromagnetic field results.

Findings

Dirac equation in Yang-Mills field reduces to a fourth-order equation for one real function.

Remaining spinor components can be made real via gauge transformation.

Extension of previous electromagnetic field results to non-Abelian gauge fields.

Abstract

Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function, and the remaining component can be made real by a gauge transformation. This work extends the result to the case of the Dirac equation in the Yang-Mills field.